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A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

Published online by Cambridge University Press:  05 October 2020

VICTOR SELIVANOV*
Affiliation:
A.P. ERSHOV INSTITUTE OF INFORMATICS SYSTEMS SB RAS AC. LAVRENTYEV AVE. 6, NOVOSIBIRSK, 630090, RUSSIAE-mail: vseliv@iis.nsk.su

Abstract

The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

Type
Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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